Topological control for min-max free boundary minimal surfaces

TL;DR

This paper establishes bounds on the topology of free boundary minimal surfaces obtained through min-max methods, revealing how topological features change in the limit within three-dimensional manifolds.

Contribution

It introduces new topological bounds and semicontinuity results for free boundary minimal surfaces derived via min-max techniques, including applications like constructing specific minimal surfaces.

Findings

Betti number is lower semicontinuous along min-max sequences

Genus decreases when boundary components increase in the limit

Constructed a free boundary minimal trinoid in the Euclidean ball

Abstract

We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.